What is the source of this halacha? One would expect it to be in the last Mishna in Tamid like the rest of the songs.

What does he mean by "Az Yashir Moshe veMi Chamocha"? If he would just say Az Yashir I would assume he means the whole shira. If he would say ad Mi Chamocha I would assume that it is the last pasuk. But what does he mean by "veMi Chamocha"?

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במוספי דשבתא מה היו אומרים? אמר רב ענן בר רבא אמר רב: הזי"ו ל"ך. ואמר רב חנן בר רבא אמר רב: כדרך שחלוקים כאן - כך חלוקין בבית הכנסת. במנחתא דשבתא מה היו אומרים? אמר רבי יוחנן: אז ישיר ומי כמוך ואז ישיר‏
At the additional sacrifice of Sabbath what did they say? — R. Anan b. Raba said in the name of Rab: "Hazyw Lak". R. Hanan b. Raba said also in the name of Rab: As these sections are divided here, so they are divided [when read on Sabbath] in the synagogue. At the afternoon sacrifice of Sabbath what did they say? — R. Johanan said: "Then sang", and "Who is like thee", and "Then sang". (Soncino)

The Gemara there continues to explain that the two sets of songs cycled in parallel, the six parts of Haazinu repeating every six weeks, and the quotes of the songs for the afternoon service every three weeks. (The Lechem Mishna wonders why Rambam omitted the third song in the afternoon from his list, but doesn't have a good answer.)